Semiparallel Isometric Immersions of 3-Dimensional Semisymmetric Riemannian Manifolds

Lumiste, Ülo

doi:10.1023/b:cmaj.0000024514.62699.e0

Cited by 5 publications

(1 citation statement)

References 14 publications

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“…In [ 5 ] the following conjecture is formulated: If a semiparallel submanifold M m in E n is intrinsically a Riemannian manifold of conullity two, then it can be only planar (according to the classification given in [ 1,6,7 ]). This conjecture arose in the study of the three-dimensional semiparallel submanifolds M 3 in E n and was confirmed for this case of m = 3 and arbitrary n in [ 5 ].…”

Section: Introductionmentioning

confidence: 58%

Semiparallel submanifolds with plane generators of codimension two in a Euclidean space

Lumiste¹

2001

Proceedings of the Estonian Academy of Sciences. Physics. Mathe

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A submanifold generated by plane leaves of codimension two in a Euclidean space is, in general, intrinsically a Riemannian manifold of conullity two. All such manifolds have been classified into four classes: planar, hyperbolic, parabolic, and elliptic, i.e. having, respectively, infinitely many, two, one, or no real intrinsically asymptotic distributions. It is proved that if such a submanifold is semiparallel and intrinsically a manifold of conullity two, then it must be planar. This verifies, for the case considered here, a conjecture that a semiparallel submanifold, which is intrinsically of conullity two, must be planar. Validity of this conjecture has been established previously by the author for the three-dimensional semiparallel submanifolds.

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Section: Introductionmentioning

confidence: 58%

Semiparallel submanifolds with plane generators of codimension two in a Euclidean space

Lumiste¹

2001

Proceedings of the Estonian Academy of Sciences. Physics. Mathe

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Classification of a class of minimal semi-Einstein submanifolds with an integrable conullity distribution

Mirzoyan¹

2008

Sb. Math.

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A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds

Chen

2019

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A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden-Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper we provide a comprehensive survey on this important class of submanifolds.

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Semiparallel Isometric Immersions of 3-Dimensional Semisymmetric Riemannian Manifolds

Cited by 5 publications

References 14 publications

Semiparallel submanifolds with plane generators of codimension two in a Euclidean space

Semiparallel submanifolds with plane generators of codimension two in a Euclidean space

Classification of a class of minimal semi-Einstein submanifolds with an integrable conullity distribution

A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds

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